Middle European Mathematical Olympiad 2010 Problem T-7

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation $$1\underbrace{0\ldots 0}_{n}2\underbrace{0\ldots 0}_{n}2\underbrace{0\ldots 0}_{n}1.$$

Prove that $a_n/3$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.